I think one of the first and key pieces of irony that we are presented with in the novel is when the Dursley family try to do everything they possibly can to escape and run away from the letters Harry keeps on receiving from Hogwarts, but all it does is make things worse! Even when they completely de-camp and move to an isolated island they cannot stop Harry receiving the letter that tells him he has been selected to study at Hogwarts as a wizard.
If you want to think about other examples of irony in the story, you can think about the way that the author plants the red herring of the villain being Snape, when all along it is professor Quiddle. The fact that Harry insists that it is Snape all along is ironic given how close he is to working out the truth on a number of occasions.
One example of irony in the first Harry Potter book is the fact that while Harry, Ron, and Hermione were fixating on Snape being the evil man trying to revive Dumbledore and at one point even pitied Quirrell for being bullied by Snape, Quirrell was the real "bad guy" at the end of the novel. This is ironic because the plot travels in almost the exact opposite direction as what the characters and readers expected. Therefore, in this scenario irony is used as a plot twist.
In the first book, Harry, Ron, and Hermione go in with the expectation that Snape is the one in league with Voldemort. In actuality, Quirrel is the one who let Voldemort occupy his body and devised a way to get to the Sorcerer's Stone. And just some light humor: Fred and George bewitch snowballs to bounce off Quirrel's Turban. That is funny in itself. But a layer of ironic humor lies in the fact that that was basically Voldemort's face.